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			<h1>Infinity!</h1>
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<p><span>		
(Copyright 1991, by Nancy Casey) 
</span></p> 
<div align="justify">
<p><span> The original document can be found at <a href="http://www.ccs3.lanl.gov/mega-math/workbk/infinity/inhotel.html">this location</a>. </span></p>
<!-- Document formatting changed by Kushal, 2010 -->
<div id="article">
 <p><span>
So!  You want to know about Sam Cantori, his hotel and those busses.
Let me guess!  You want to know everything---all the exactly's and the
how-so's.   You only want the truth, you say.
The whole truth---complete and exact.  I can tell it to you. 

 
 </span></p>  <p><span>
A warning to you, though:
if you only want to hear things that
are easy to believe, why don't you just tell me now and save me the
trouble of wasting my breath.
And if you are the type that likes to say
words like ''impossible'', or if you think you are going to interrupt
and try to tell me that things I saw with these two eyes couldn't have
happened, or things that I made with these two hands couldn't have
been built---well, why don't you just get up right now and go swap
stories with Mr. Kronecker.  And when you get there, don't let him say
a word until he tells you how he learned to talk.  Ha! 
 
 </span></p> <p><span>
But if you want to listen, I'm glad to tell you all the stories I
know, and if you help me, I'll try to remember the rest.
 If we can get all the remembering done, maybe you
could build it all back up again.  Then you could have some of the
fun---because, oh, it was fun we had at the Hotel Infinity. 
 
 </span></p> <p><span>
It was a long time ago that Sam hired me.  I think he picked me
because I had already lived and worked a long time and could do many
things.  He didn't have to teach me how to build a hallway, or a room,
or how to put in pipes for water, or wires for electricity.  I could
build with metal just as easily as with wood, and had worked on motors
and engines since I was 9, so there was nothing in the putting
together of the building or the busses that I couldn't do. 
 
 </span></p> <p><span>
Being on the job with Sam was different than with any other boss I've had.
I'd get a staircase or a hallway all framed in the way he said, then he'd
check the blueprints he kept locked in the safe, and have me make tiny
changes---a little twist for the steps, a slant on the ceiling, a bit
of
lean on the wall.  None of his changes made any sense for anything except
making a good building bad.  But I did what he said.  I did what he told
me.  None of this ''Why do I have to do it this way, boss?'' or ''I
think I
know a better way, boss,'' or ''This doesn't make any sense to me,
boss.''
None of that.  It was all ''Yes, sir,'' and ''Right away, sir,'' and
''What next, sir?'' 
 
 </span></p> <p><span>
Sam Cantori paid me to do, he didn't pay me to think.  Once I started
thinking about what it was I had done, it came to me that I had done
some things that were downright impossible, things I'd stand here
today and tell you couldn't be done, except that I did 'em myself. 
 
 </span></p> <p><span>
There didn't seem to be anything orderly about the way we worked
either.  One day it was floors, doorknobs, and fan belts.  Next day
he'd put me to working on transmissions, sinks, and wallpaper.  If he
tilted a mirror a little crooked, or had me raise one end of a step,
he might then say, ''Now go do that to all the rest of the busses and
in all the rooms while I study the plans and decide what to do next.'' 
 
 </span></p> <p><span>
All the time I worked, the questions I never asked built up in me
somehow, because I remember the morning when I started to be
different.  I showed up for work, right on time like I always do,
carrying my tools.  And there was Sam in his office.  All the
blueprints were gone, all the plans, all the lists, all the sawdust,
the nails, the screws, the paint, the grease, the sawhorses, the
gears.  The floor was swept clean as could be, and what a sight was
Sam!  His slick and shiny hair showed the tracks that the comb had
left.  His chin was clean-shaven, pink and smooth.  The suit he had on
was made of the finest cloth I'd ever seen or touched, and a cane
dangled from his wrist.  In one hand was a silk top hat, in the other
a gleaming gold watch.  He snapped the watch case shut, tucked it into
his vest pocket and looked up at me as I came in. 
 
 </span></p> <p><span>
''Sam!'' I cried, ''What's---'' 
 
 </span></p> <p><span>
He held up his white-gloved hand for silence, and motioned me over to
his desk.  He shoved two papers in my direction.  They were drawings. 
 
 </span></p> <p><span>
''Make these two signs, George.  When you've got them hung up, put
your tools on the pile with the other things out back.  I'm going to
call Kronecker to haul all that junk away.  Then go to the tailor, the
cobbler, and the barber.  They're expecting you.  No more of this
backbreaking sweat and sawdust.  You're the one who will greet all the
guests at the Hotel Infinity!'' 
 
 </span></p> <p><span>
The moment I saw those drawings, something changed in me, and I turned
into a questioning, attention-paying person.  Oh, I didn't speak any
questions out loud, but it was the first time an idea came into my
head about not doing something exactly the same as Sam Cantori told me
to do it.  ''Make these two signs and we're done,'' he'd said.
Impossible! 
 
 </span></p> <p><span>
One sign was the big one to go out front:   "Hotel Infinity",
it
said.  "All rooms One Dollar".  That's a strange enough name, and
a price sure to lose money, but that wasn't the problem.  Sam was rich
and only doing this for fun anyway, and he could name his hotel
anything he wanted.  It was the other sign, the one that said
" Vacancy" that bothered me.  Not too strange, just a little bit
strange, like those crooked walls and bending hallways.  But this was
a different morning.  For some reason I was thinking about things. 
 
 </span></p> <p><span>
Now there's nothing wrong with a "Vacancy" sign that hangs under
the name of your hotel to let people know there's room for them.  But
everybody knows that you have to have a little sign that says, "No" to 
hang up in front of the "Vacancy" sign so it says "No Vacancy" when the 
hotel gets full.  You get in trouble if you
advertise a vacancy, then tell someone you're all full up and they
can't come in.  I started to bring it up with Sam, but he was already
on the phone with the junk man Kronecker negotiating a price for our
big pile of leftover materials and tools.  So I made that little sign,
too, and hid it under the pot of the avocado tree.  Here it is.  As
you can see, we never hung it up. 
 
 </span></p> <p><span>
When I came back from getting myself spruced up like Sam, Kronecker
was leaning against the handle bars of that propeller-driven
solar-powered truck he made out of hot water tanks and bicycle parts,
staring at the big sign in the parking lot.  I knew he was going to be
glad to get that junk pile.  He's a funny man, that Kronecker.  He
won't use anything he doesn't make himself out of bits and pieces of
everyday things, so all that junk was a pile of treasure for him.
That's why I was surprised when he whirled and shook a bony finger at
me. 
 
 </span></p> <p><span>
''Cantori told me what you two were up to,'' he growled.  ''Hotel
Infinity,
indeed.  No such place can be made, yet you pretend that it can.  Such a
thing is incorrect, evil, and wrong.  I will see to it that this is
stopped.  And yes, I WILL take your tools, because you two
should never be allowed to claim to build another thing!'' 
 
 </span></p> <p><span>
Sure, Kronecker's threat was strange, but Kronecker was strange.  There was
no time to argue or worry about what he said.  The first guests were
already arriving.  It's no surprise that the place was already beginning to
fill up.  Where else can you get a hotel room for a dollar? 
 
 </span></p> <p><span>
Sam stood up from his big leather chair and greeted the members of the
first family in line.  ''What is it you like?  What do you like the
best?'' he boomed, pumping the hand of the youngest child. 
 
 </span></p> <p><span>
''Purple!'' came the answer---along with a grin that had one tooth
missing. 
 
 </span></p> <p><span>
''And you?"  He reached for the hand of the older child and gave it a
big shake.  ''What do you want in your room?'' 
 
 </span></p> <p><span>
''Polka dots!'' 
 
 </span></p> <p><span>
''Purple with polka dots.  Hmmm.  Will those be orange polka dots or
crimson ones?'' he asked the parents. 
 
 </span></p> <p><span>
Theh parents turned to each other and shrugged,
''Orange will be just fine,'' they said. 
 
 </span></p> <p><span>
''Wonderful!'' clapped Sam.  ''Room 91 is the one for you.  I'll take
your dollar.  Then all you need to do is read this very important
paper here.  Grownups, please make sure the kids understand.
And
when you've signed it, my assistant George will show you how to find
your room." 
 
 </span></p> <p><span>
Now, that paper was a funny thing.  Sam always took charge of that.
The paper had all the usual things on it, you know, clean up your
mess, fix what you break, check-out time is 11AM.  But there was this
one extra thing at the bottom that Sam always acted like it was so
important.  It was written in letters twice as big as the rest, and
Sam always made sure that they saw it.  It said,
''AND I PROMISE TO OBEY EVERY REQUEST THAT IS MADE OVER THE
LOUDSPEAKER, NO MATTER HOW SILLY OR INCONVENIENT IT MIGHT SEEM.''  
 
 </span></p> <p><span>
Well, the silly part was that
we
didn't ever use the loudspeaker, even though I knew full well that it
went everywhere in the whole place.  I put it there myself, you know. 
 
 </span></p> <p><span>
Well, my job was to show the people to their rooms.  I didn't have to
go all the way there.  The place was too huge for that.  Sam would
hand me a piece of paper with their room number on it.  I would tell
them the number and start them on their way.  When we were just new
and people were just starting to find out about us, I told them
numbers like 6, and 28, and 42.  In a while, as the hotel got more
popular, I was giving out more interesting numbers like 523, or 954,
and even 1,255.  After a long time, I got used to handing out numbers
like, 3,244, and 8,999,223.  It was all very organized the way the
hotel was set up, so even though the numbers got very large, the rooms
were easy to find.  I liked the big numbers the best. 
 
 </span></p> <p><span>
One day, when I had just sent a family off to room 4,693,203,355,226 I
asked Sam the first question ever.  ''How many rooms do we have in
this place, anyway?'' He looked at me kind of funny.  ''Just curious,
sir,'' I said quickly. 
 
 </span></p> <p><span>
''You built it, George, you should know,'' he answered.  ''Infinitely
many.'' I didn't ask any more questions for a long time.  But I did
watch, and I did think. 
 
 </span></p> <p><span>
One day, Sam said he would be out for a while.  He had an appointment
with the head of a tour company that was probably going to buy the
busses for sure.  He said that I was to be in charge while he was
gone.  I would be the one who sits at the big desk and calls a big
welcome to everyone who comes in the door:  the families, the little
children and their pets, the salespeople, the convention-goers, the
bicycle travelers, the bus loads of tourists.  I couldn't wait. 
 
 </span></p> <p><span>
''Make sure they understand that paper,'' Sam cautioned.  ''Don't let
anyone in without signing it.  And oh, by the way--there's only one
room empty right now.  It's 26.'' He gave a wave and was jauntily off,
swinging his cane. 
 
 </span></p> <p><span>
Only one empty room!  Thank goodness I had made that sign.  It was
going to have to say "No Vacancy"  out there before this day was
over for sure!  I kind of wondered how many people actually were in
the hotel just now.  That would be interesting to know.  But the
office was filling up with folks like it always did. 
 
 </span></p> <p><span>
At the same time that cars were pulling up in the lot outside, a few
people were dragging in suitcases to leave.  A line had already formed
at the right-hand side of the desk where the sign said, "Check
In".  I had to get that sign up fast.  When I tried to tip the pot of
the avocado tree to pull the little "No" sign out, it wouldn't
budge.  The tree had grown too huge and heavy since I slipped it under
there.  I was going to need a lever to get the sign out, and a hammer
to hang it up.  But Kronecker had taken away all the tools years ago.
It was useless.  What was I to do?  I had to think. 
 
 </span></p> <p><span>
As quickly as I began to think, I had an idea.  It wouldn't work for
sure, but it would buy me some time.  ''Good morning,'' I said to the
first person in the check-in line.  He wore a suit and carried a
briefcase.  I gave him the paper, made sure he agreed to all of the
rules, including the one about the loudspeaker, took his dollar, and
sent him off to room 26.  Next in line were two young kids wearing
backpacks. 
 
 </span></p> <p><span>
''We're traveling by bicycle.'' they said. ''Usually we camp out, but
we heard about this place.
The Hotel Infinity.  History making.  We could use a shower and a soft bed,
so we decided to stay here tonight.  Here's our dollar.'' 
 
 </span></p> <p><span>
''Just a minute,'' I said, and swung myself around in Sam's big
leather
chair to face the first person in line in front of the left-hand side of
the desk where the sign said "Check Out". The woman handed me the
key to Room 4,326. 
 
 </span></p> <p><span>
''Was everything okay,'' I asked her. 
 
 </span></p> <p><span>
''Oh, it was just fine,'' she said.  ''I've never slept so well in a
bed shaped like a turtle before.'' She picked up her suitcase and
walked
toward the door.  The man behind her started to hand me a key, but I
held up my hand, smiled and said, ''Be with you in just a minute,
Sir.'' Quickly, I swung the chair back around to face the bike
trip kids and checked them into room 4,326.  The man who I'd left
waiting checked out of room 4, which is where I checked the next
person into.  That's the way it went all morning long.  I kept
checking people into whichever room it was that someone just checked
out of.  I was lucky.  After the last person checked out, there was
exactly one person left to check in.  What a relief! 
 
 </span></p> <p><span>
The lobby was empty.  I leaned back into the soft leather of Sam's
chair, propped my feet on the desk and looked at my reflection in the
polished shine of my fancy shoes.  I was proud of myself.  Just a
little bit of thinking, and I had figured out what to do.  Outside, I
could hear splashing in the pool, the clop-clop of the ponies feet,
and the music from the carousel.  People were playing tennis,
strolling on the sidewalks, pitching horseshoes, shooting arrows
at the archery targets, and sipping cool
drinks.  A perfect day at the Hotel Infinity.  Then, right there
before my eyes, a car pulled up.  Oh no!  Nobody would be checking out
until tomorrow, and I had forgotten to put up the sign. 
 
 </span></p> <p><span>
Every door of the car creaked slowly open, and a whole tired-looking
family got out, along with two cats, a Saint Bernard, and a poodle.
The girl was carrying a bird cage, and her brother had a fish
bowl.  The man pushed the office door open and tossed a crumpled
dollar bill on the desk.  He didn't look friendly. 
 
 </span></p> <p><span>
''We need a room,'' he gruffed.  ''A big one.  Now.  Been driving
since yesterday morning.'' He held his hand out for a key.  ''Which
one is it?'' 
 
 </span></p> <p><span>
I started to explain that I didn't have a room, but he interrupted me.
He threatened me and scared me.  He told me I was going to go to jail
for running a hotel that advertised a vacancy, and refusing to give
him a room.  He said I was discriminating against families, against
tired people, against loud people, against rude people, against people
who had been driving since yesterday morning.  He grabbed the phone
and was calling his lawyer, when in burst Sam. 
 
 </span></p> <p><span>
''George!  I've done it!'' he shouted.  ''I sold them!  I've sold all
the busses to Continue 'Em Tours.  All of them, George!  Sold!  It's a
wonderful day, George!'' 
 
 </span></p> <p><span>
''Oh, Sam, Sam, no.  It's a horrible day,'' I groaned.  ''Forget the
busses, we're ruined.  Something terrible has happened.'' Sam's face
fell, and I started to explain.  ''The hotel is full, Sam, and there's
no place to put this man.  He's mad, he's going to sue.  We're going
to jail!  Oh Sam, it's so terrible, Sam, I should have told you
sooner, we need to change the sign. . .'' 
 
 </span></p> <p><span>
Sam wasn't even listening.  He strolled over to the man and said,
''Excuse me, Sir, my untrained assistant has made a mistake.  He meant
no harm, he just doesn't know how to find an empty room when the hotel
is full.  It is true that we already have an infinite number of guests
but this is the Hotel Infinity.  We have an infinite number of rooms,
so of course there is space for you.  Please, have a seat, this will
only take a moment." He bustled over to me, fishing into his vest
pocket for a piece of paper which he pressed into my hand. 
 
 </span></p> <p><span>
''At last, George,'' he whispered, grinning.  ''We get to use the
loudspeaker.  Go announce this.'' 
 
 </span></p> <p><span>
I went over to the table in the corner with the microphone on
it, angry and muttering all the while.  ''Untrained assistant!
Haven't
I worked for Sam more years than the two of us put together can
remember?  Finding an empty room when the hotel is full.  An infinite
number of guests.  An infinite number of rooms.  An infinite number of
too-crazy things is what we've got here!'' I flipped the switch on
the loudspeaker system, and much to my surprise and relief, it worked
after all these years.  I leaned over the microphone and read what was
on the paper. 
 
 </span></p> <p><span>
''Ladies and gentlemen, families and friends.  Please recall that you
have agreed to do exactly as we ask when we make a request on the
loudspeaker, even if what we ask seems silly or inconvenient.  This
won't be hard and it won't take long.  All we ask is this:  would
everyone please gather up your belongings and move to the room whose
number is exactly one greater than the number you are in.  Got that?
Everyone move quickly now, yes that's right, to the room whose number
is one greater than the one you are in.'' 
 
 </span></p> <p><span>
The people had signed.  The people had agreed.  So, although the
people were somewhat confused, they began to move.  The trapdoor in
the ceiling of room 34 creaked open, and that family climbed out and
moved on to room 35.  The businessman in room 578 appeared in the
hallway and asked a boy where room 579 was. 
 
 </span></p> <p><span>
''Oh, my mom and I just left there,'' he said, ''it's back that way
just around the bend in the hallway.  We're looking for room 580
now.'' 
 
 </span></p> <p><span>
The third graders on a class trip quickly gathered their things and moved
out rooms 222, 223, and 224 and into to rooms 223, 224, and 225.  The
grandparents in room 3,465,892 moved to room 3,465,893.  And the lonely old
man who had been in room 1 did not look nearly so glum as he stood on
the ladder, pushed up on the round door over his head, then climbed
up and popped through the floor into to room 2. 
 
 </span></p> <p><span>
''Well!'' said Sam, as he turned to the man, who still looked rather
angry.
''We are most sorry for the inconvenience, sir.  So sorry.  As you can
imagine, it has taken many years for this hotel to get full.  This is the
first time it has happened, and my assistant here wasn't aware of the
procedures.  Please accept my apologies, and let me return your dollar so
that you can stay in room number 1 for free.  But first, of course, you
must sign the paper, here.'' 
 
 </span></p> <p><span>
That Sam!  He was such a charmer!  Oh, those were the days---before
all that fuss with Kronecker heated up. 
 
 </span></p> <p><span>
From then on, more people came than left every day, so the hotel was always
full.  We used the loudspeaker to get everyone to move down whenever we
needed an extra room.  It got to be known far and wide that this would
happen, and people began coming to the hotel just to see how many different
rooms they could get to stay in.  They might start out in an upstairs room
with dinosaurs on the wall paper, and then the loudspeaker call would come,
and they would have to move to the room with the checkerboard
bedspreads.
If it was a particularly busy day, a lucky family might, all in one day,
get to stay first in the room with the elephant-shaped bathtub, then in
the one with jungle plants and trapezes.  After dinner they might get sent
to the room that has the bathtub slide, and end up going to sleep in the
one with the fold-down beds.  People didn't mind moving.  That
turned out to be half the fun. 
 
 </span></p> <p><span>
Because I was paying more attention, I was learning to manage the
hotel.  Sam could leave me in charge, and I could handle everything
just as smoothly as he could.  Good thing, too, because Sam had to be
away more and more due to all the trouble Kronecker was causing. 
 
 </span></p> <p><span>
I think it was Kronecker's peculiarity about having to know how things
were built up from simple parts that set him off.  And Sam is pretty
stubborn, too.  He said that if Kronecker wanted to be a fool and not
believe the hotel existed just because he didn't know how it was made,
that was up to him.  Then Kronecker bought up all the newspapers and
wouldn't run Sam's ads because he said they violated
truth-in-advertising laws.  That's why they ended up in court.  Lots
of people from the hotel came to see, but of course they didn't all
fit in the courtroom. 
 
 </span></p> <p><span>
Sam stood up in front of the judge and declared, ''At this moment
there are infinitely many people inside the building that Kronecker
claims everyone should ignore." He pounded his fist on the witness
stand.  ''He has never set foot in the door!  If he did come inside,
see the people come and go, try and count them if he likes, and watch
our operation, he would see how it works.  It is indeed infinite.
Why, I'm sure we could find room for any number of new guests right
now.'' 
 
 </span></p> <p><span>
''Preposterous!'' shouted Kronecker ''He tells me to come visit the
infinite
as though it is something waiting to be discovered or explored, skipping
over entirely the fact that ordinary people can not figure out a way for it
to get there.  If he cannot tell me how that place is constructed, it is
neither real nor useful and people should stay away!  And
furthermore---'' 
 
 </span></p> <p><span>
''Listen to this ridiculous man!'' burst in Sam.  ''I am sure he
cannot tell
me how to construct a stomach, yet this has never prevented him from
staying away from his lunch!'' 
 
 </span></p> <p><span>
There was a lot of shouting back and forth like that, and the judge
kept having to pound his gavel and make them calm down.  In the end,
though, he pretty much sided with Sam when he said, ''He claims that
there are infinitely many people inside, and that there is still room
for more.  I agree, Mr.  Kronecker, that this is quite puzzling, but
just because something is puzzling, we can't be telling people it is
dangerous and they should stay away.  There are ways, Mr.  Kronecker,
that you could prove that what Mr.  Cantori claims about his hotel is
not true.  If there aren't infinitely many people there, as you claim,
just count them and tell us how many there are really.  And it also
seems to me that if someone is turned away because the hotel is full,
you would have proved your point about Mr.  Cantori's advertising
being false.  If you can present the court with this kind of
information, we will reopen this case, but for now I rule that you
must allow Mr.  Cantori to advertise.  Case Dismissed!" 
 
 </span></p> <p><span>
Sam laughed and looked at Kronecker, ''Come on over and count.  Or
come on over and wait for us to turn a guest away.  You will be at
either task for infinitely many days!'' He paused, and scratched his
head.  His voice got quieter.  ''Or maybe a little longer.  I'm not
sure.  I'll have to think about that.'' 
 
 </span></p> <p><span>
Kronecker shook his bony finger and threatened revenge, but things
settled down.  A good thing, too, because I don't know what I'd have
done if I'd been there alone the day that one of Sam's busses pulled
up---fully loaded and needing rooms for everyone. 
 
 </span></p> <p><span>
I told you I made those busses just like I made the hotel, back when
all I did was follow orders, before I started asking questions.  Did
I tell you that those busses were made just like the hotel?  That the
busses held infinitely many people?  Sam saw it pull up before I did.
He rushed right out and began talking to the driver.  I knew it would
be time for my sign for sure.  I hoped the driver wouldn't be too
mad---or the people on that bus either.  Where else is a bus
loaded
with infinitely many people ever going to be able to let all the
passengers off so they can take a shower and sleep in a nice warm bed?
Oh, I hoped Kronecker wouldn't get wind of this. 
 
 </span></p> <p><span>
Well, wasn't I surprised when Sam Cantori shook the driver's hand and
started waving the people off the bus.  As he walked inside he was
fishing into his vest pocket with his eyes all bright.  He handed me
a note, and flashed me a grin.  ''Loudspeaker time, George.  By Zeno!
I'm glad you made those busses.  I don't see how we'd have ever gotten
to this any other way!'' 
 
 </span></p> <p><span>
The lobby was filling up with people, and more and more of them were
pouring out of the bus.  No time to study the note---I just went
to the loudspeaker and read it off. 
 
 </span></p> <p><span>
''Ladies and gentlemen, families and friends.  Recall that you agreed
that you would follow our requests on the loudspeaker, no matter how
silly or inconvenient they seem.  Here is what we must ask you to do.
Please look on the outside of your door and copy your room number down
on a piece of paper.  Everybody got that?  Now, everyone, please
multiply that number by 2.  Yes, double the number, and write down
your answer.  Check your work please, make sure that it is correct.
That's your new room number, folks.  If you would please, gather up
your things as quickly as you can, and move into the room that has
your new number.'' 
 
 </span></p> <p><span>
And they did.  The person in room 1 moved to room 2.  The family in
room 2 moved to room 4, the kids in room 3 moved to room six, and so
on in all the rooms.  The room 4,886 was now occupied by the people
that had been in room 2,443. The driver was motioning people off
the bus, there were people everywhere, infinitely many people!  Where
would we put them all? 
 
 </span></p> <p><span>
Sam pulled me aside and whispered, ''It's OK, George, I told them they
could take any odd-numbered room that they wanted.  There are
infinitely many odd numbers, you know, so it should come out just
right, don't you think?'' 
 
 </span></p> <p><span>
Well, it did come out just right.  And people kept coming.  For years
and years they came to see and visit and change rooms at the Hotel
Infinity.  Sometimes it was a carload of people that would come,
sometimes just one person on roller skates or on a unicycle, sometimes
a regular bus, and sometimes one of Sam's busses for infinitely many
people.  It always worked.  Once five of the Continue 'Em Tour busses
came at once.  Another time there were 10.  And one time over a
hundred and fifty of those bus loads pulled up at once.  No problem!
We could always fit them in.  Until the day when---well, you've
probably all heard plenty about that. 
 
 </span></p> <p><span>
Some say that Kronecker had a hand in the disaster.  Maybe he did, but
you want the truth, so I can only tell you what I know.  He was the
one who rented all of the Continue 'Em Tour Busses that day, filled
them up and sent them over.  I know that because the Continue 'Em Tour
director called and told us.  She knew all about the hotel, of course,
but even she was a little concerned about what could happen if all of
the busses showed up at once.  I told her not to worry, and passed the
message along to Sam just to let him know, and to my surprise, he
looked alarmed! 
 
 </span></p> <p><span>
He hurried to the safe and unlocked it for the first time in
oh-so-many years.  The crinkled papers he pulled out were covered with
mathematical-looking writing. 
 
 </span></p> <p><span>
''When I discovered this trick, I was amazed that such a thing could
be possible,'' he said.  ''Now, how does it go again?'' 
 
 </span></p> <p><span>
I could hear rumbling in the parking lot.  The busses were arriving.  He
began to shuffle faster and faster through the brittle pages, and as
he did, they crumbled into hundreds, thousands, maybe infinitely many
flakes of old yellowed paper. 
 
 </span></p> <p><span>
''It's lost!'' he cried out.  ''The secret is lost!  If I cannot
remember, Kronecker will have beaten us,'' and he buried his head in
his hands. 
 
 </span></p> <p><span>
''Sam,'' I said, ''It cannot be so bad.  We have accommodated many
busses before.  We will just do the usual trick.  The people like the
loudspeaker.  They don't mind moving.  So what if it is 200 or 500 or
even a thousand busses.  How many of those busses did we build
anyway?'' 
 
 </span></p> <p><span>
He looked up at me in a way he never had before.  He spoke to me, not like
I was his assistant, his builder, or his employee.  Sam
spoke to me like the old friend he had become. 
 
 </span></p> <p><span>
He said my name quietly.  ''George, you don't know, do you?'' 
 
 </span></p> <p><span>
I shook my head.  I had stopped trying to count things so long ago. Of
course I didn't know. 
 
 </span></p> <p><span>
''George, I have maybe done a terrible thing keeping it all so secret,
secret even from you.  I left it up to you to notice, George, but I didn't
really think you would.  You didn't notice, did you, that every time we
finished a room in the hotel we finished a bus.'' 
 
 </span></p> <p><span>
I stared at him. ''A bus for every room?''  No, I hadn't noticed. 
 
 </span></p> <p><span>
''Yes,'' he said.  ''As many busses as there are rooms.'' 
 
 </span></p> <p><span>
''Then that means...'' 
 
 </span></p> <p><span>
''Yes,'' he said.  ''Infinitely many busses.  And they are beginning
to line up outside now.  I cannot remember what to do to fit all those
people in.  We are finished.  Let Kronecker say he has beaten us.'' He
stood up and hobbled over to the avocado tree.  ''I need your help,''
he said, slipping the brass tip of his cane under the lip of the pot.
We leaned hard together and the heavy tree tipped a bit.  With a
deft kick of his once-shiny shoe, he sent that little sign I had
made skittering out across the floor. 
 
 </span></p> <p><span>
''I'd always been amused by your little secret,'' he smiled sadly.
''Go ahead.  Put it up.  Let the Hotel Infinity say 'No Vacancy'.
Kronecker can take a picture for all his newspapers.'' 
 
 </span></p> <p><span>
Next to that potted tree, he stood, not straight and tall in the
manner of the Sam I will always remember, but hunched over, his fine
clothes hanging loosely from the bony hump between his shoulders.  He
turned and went through door number 8.  The draught from the hallway
that leads to all the rooms whose numbers begin with '8' caught the
blueprints and they floated like giant feathers to the floor.  The
fragments of crinkled paper whirled and fluttered like snow.  The
last I saw
before the door closed was a slow-moving figure that didn't carry his
cane over his wrist in that old jaunty way.  He used it to support
himself as he hobbled along. 
 
 </span></p> <p><span>
''Sam is old,'' I said to myself.  The door clicked shut behind him,
and the
nail that fastened the wooden '8' to the door must have come loose, because
the number tipped over on its side.  ''This hotel is old, too,'' I
said. ''So am I.  But I'm not too old to think.'' 
 
 </span></p> <p><span>
There was honking and shouting in the parking lot.  The drivers who
were awaiting their instructions for unloading the busses were getting
impatient, and more busses were pulling up.  I had disobeyed Sam when
I made that little sign, and I was ready to disobey him again.
  First I made the usual announcement that emptied out all
the odd-numbered rooms.  That gave me some time to think. 
 
 </span></p> <p><span>
''Infinitely many odd numbers,'' I said to myself.  ''Infinitely many
rooms empty, I know that much for sure.  How can there be more than
infinitely many people on those busses?'' 
 
 </span></p> <p><span>
Then, I started out slowly toward that parking lot, and with every
step I thought harder and deeper.  When I got to the first bus, I had
an idea that was worth a try. 
 
 </span></p> <p><span>
I told the first couple of drivers to pass the word down that all the
people would file off the busses and fill up the odd-numbered rooms.
Now all I had to do was get all the folks to move off the busses in an
orderly way so no one would get skipped, and no one would have
infinitely many people in line in front of them.  Then they could fill
the odd numbered rooms in order. 
 
 </span></p> <p><span>
''OK,'' I said.  ''Now, we'll take the first person off the first bus
and the first person off the second bus.'' A lady with a briefcase.  A
man and his dog.  No problem.  ''Now, we get the next two people off
the first bus, one person from the second bus, one person from the
third bus, and one person from the fourth bus.'' 
 
 </span></p> <p><span>
You know, people who come to the Hotel Infinity are all so wonderful.
Other folks might have minded the wait, but these people were
cheerful, like it was a game.  People who came to the Hotel Infinity
counted on doing something unusual. 
 
 </span></p> <p><span>
After the first person got off the fourth bus, we worked our way
backwards, taking one person off the third bus, the second bus, and
the first bus.  Then one more off the first bus, and we worked our way
back up, taking one person from each bus that was already in the game,
and adding a person from the fifth bus and a person from the sixth bus
this time.  Back and forth we went, always letting two people off the
first bus, and starting to empty two more busses each round. 
 
 </span></p> <p><span>
It looked like it would work.  More and more people were moving.  At
each round, more busses came into the game and the people began to
file out.  The odd numbered rooms in the hotel were slowly filling up.
Word spread down from driver to driver about how the busses were being
unloaded.  You could hear whispering in the busses as the people
passed the word to one another about how it was all working. 
 
 </span></p> <p><span>
At least I think it was all working.  I think it would have worked.
After they didn't need my help on the busses anymore I went inside and
sat in the big leather chair.  The rooms in the seven hundreds were
about half full when people started rushing out saying they smelled
smoke.  I didn't wait to sound the alarm.  A big wooden building that
old, and all those nice people---I couldn't put them in danger.  I got
on the loudspeaker and told everyone to get back on the
busses--including the people in the even-numbered rooms.  The guests
were all out and the busses were pulling away for safety when the big
blaze started to roar. 
 
 </span></p> <p><span>
Maybe the building was just too old and it was hopeless.  Maybe the
fire fighters could have saved it if it had been made in a different
way.  You know there were all those twists and slants I told you
about.  Ever so slight, but they mattered.  They only thought they
could head down Hallway Eight to get around the other side of Hallway
Seven where the fire seemed to have started.  Of course it seems like
it ought to go together that way, but nothing fit together inside that
building the way you'd think it would fit.  I guess that's the reason
why it all fit. 
 
 </span></p> <p><span>
Anyway, it's good they went down Hallway Eight instead of Hallway Two
like they should have.  Because they found Sam, hunched and hobbling,
rounding the curve towards the eight millions.  Oh, he gave him a
time, he did, holding them at bay with his cane and hollering at them
to get out, that there was no vacancy here, and that the Hotel
Infinity couldn't be on fire because there WAS  no Hotel Infinity
anymore.  They took him to the hospital figuring he was in shock about
the fire. 
 
 </span></p> <p><span>
Well before long it was true.  There was no Hotel Infinity
anymore, just hot ashes, coals, and thin plumes of smoke.  I've been
thinking since then that if we could have saved just one hallway we'd
still be in business.  But we didn't.  All that's left is the sign
that stood up outside, and this little sign here that I never put down
after Sam handed it to me.  I'm going down to the hospital to try and
see Sam.  I go every day about this time.  So far, though, he refuses
to see anyone.  If he'll talk to me today I'm going to ask him if
maybe it would be OK if I put the little sign up now. 
 
 </span></p> <p><span>
"Hotel Infinity, No Vacancy." People will see it and wonder if
that's possible, wonder how such a thing could be true.  
 </span></p> 
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